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%I #22 Sep 06 2023 13:24:47
%S 1,1,0,1,2,0,1,0,1,0,1,2,1,0,0,1,0,2,2,0,0,1,2,1,0,1,0,0,1,0,1,4,1,0,
%T 0,0,1,2,2,0,3,2,0,0,0,1,0,1,4,2,0,1,0,0,0,1,2,1,0,3,6,1,0,0,0,0,1,0,
%U 2,4,3,0,4,2,0,0,0,0,1,2,1,0,3,8,3,0,1,0,0,0,0
%N Triangle read by rows: T(n,k) is the number of compositions of n with k parts and differences all equal to 1 or -1.
%C Parts will alternate between being odd and even. For even k, a composition cannot be the same as its reversal and therefore for even k, T(n,k) is even.
%H Alois P. Heinz, <a href="/A309938/b309938.txt">Rows n = 1..200, flattened</a>
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 0, 1, 0;
%e 1, 2, 1, 0, 0;
%e 1, 0, 2, 2, 0, 0;
%e 1, 2, 1, 0, 1, 0, 0;
%e 1, 0, 1, 4, 1, 0, 0, 0;
%e 1, 2, 2, 0, 3, 2, 0, 0, 0;
%e 1, 0, 1, 4, 2, 0, 1, 0, 0, 0;
%e 1, 2, 1, 0, 3, 6, 1, 0, 0, 0, 0;
%e 1, 0, 2, 4, 3, 0, 4, 2, 0, 0, 0, 0;
%e 1, 2, 1, 0, 3, 8, 3, 0, 1, 0, 0, 0, 0;
%e 1, 0, 1, 4, 3, 0, 6, 8, 1, 0, 0, 0, 0, 0;
%e 1, 2, 2, 0, 4, 10, 5, 0, 5, 2, 0, 0, 0, 0, 0;
%e ...
%e For n = 6 there are a total of 5 compositions:
%e k = 1: (6)
%e k = 3: (123), (321)
%e k = 4: (2121), (1212)
%p b:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
%p `if`(n=i, x, add(expand(x*b(n-i, i+j)), j=[-1, 1])))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(b(n, j), j=1..n)):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Jul 22 2023
%t b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, x, Sum[Expand[x*b[n - i, i + j]], {j, {-1, 1}}]]];
%t T[n_] := CoefficientList[Sum[b[n, j], {j, 1, n}], x] // Rest // PadRight[#, n]&;
%t Table[T[n], {n, 1, 13}] // Flatten (* _Jean-François Alcover_, Sep 06 2023, after _Alois P. Heinz_ *)
%o (PARI)
%o step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
%o T(n)={my(v=vector(n), R=matid(n), m=0); while(R, m++; v[m]+=vecsum(R[n,]); R=step(R,n)); v}
%o for(n=1, 15, print(T(n)))
%Y Row sums are A173258.
%Y T(2n,n) gives A364529.
%Y Cf. A309931, A309937, A309939, A325557.
%K nonn,tabl
%O 1,5
%A _Andrew Howroyd_, Aug 23 2019
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